The magnetostatic field of a periodic cylindrical array of perfect conductors of arbitrary xy cross section
Abstract
The magnetic field in an imploding wire array is studied. After the initial explosion of the wires the array starts imploding. The process can be considered quasistatic electromagnetically and the assumption of perfect conduction is made. Under these conditions the driving magnetostatic forces generated by the wires need to be calculated in a series of snapshots, which are stages of the magnetohydrodynamics process. For that purpose a Green's function technique is developed to solve for the magnetostatic field generated by an array of perfectly conducting wires of arbitrary xy cross section, carrying current in the z direction. The wires are enclosed in an outer cylinder, inside of which they are arranged with angular periodicity. This outer cylinder carries all the returning current, and ideally does not permit any magnetic lines to escape from its enclosure. The resulting equations for the magnetic vector potential (A=A?,∇^{2}A=0 in vacuum, A=0 on the outer cylinder, A=const≠0 on the inner conductors) are implicitly solved by writing the potential as a linear superposition of point (z lines) current densities around the boundaries of the inner conductors. The point current densities are determined by imposition of the boundary conditions. For the case of small round wires not overlapping and sufficiently far from the center, an explicit solution is found in the form of an angular Fourier series, having as expansion parameter the ratio between the radius of the inner conductors and the distance to the center, times the number of wires periodically arranged. For the general case the problem is discretized by considering the boundary of each of the inner conductors as a connected set of segments of either circumferences or straight lines, each segment carrying a different amount of current. This procedure yields a set of N linear equations with N unknowns; the N unknowns are the currents carried by each of the N segments by which the boundary is approximated. Knowledge of these currents solves the problem completely because with them and using the Green's function and its complex derivative, the vector potential and magnetic field, respectively, can be obtained anywhere in the region of interest. In particular the magnitude of the magnetic field just on the boundary of each of the inner conductors is related to the magnitude of the segment currents through a proportionality constant. An algorithm to solve the linear equations is discussed and representative cases shown. The method is seen to be accurate and fast.
 Publication:

Journal of Applied Physics
 Pub Date:
 January 1979
 DOI:
 10.1063/1.325707
 Bibcode:
 1979JAP....50...23W
 Keywords:

 Computer Aided Design;
 Electric Wire;
 Green'S Functions;
 Magnetohydrodynamic Flow;
 Magnetostatic Fields;
 Series Expansion;
 Arrays;
 Boundary Value Problems;
 Circular Cylinders;
 Current Distribution;
 Subroutines;
 Time Response;
 Plasma Physics;
 41.10.Dq;
 52.30.+r;
 02.60.Lj;
 Ordinary and partial differential equations;
 boundary value problems